# Deriving |𝐶𝐶𝑍⟩ magic states from |𝐶𝐶𝐶𝑍⟩?

Analogous to the $$|T\rangle$$ and $$|CCZ\rangle$$ magic states, one can define a $$|C^n Z\rangle$$ magic state. Is there any known quantification of the amount of magic of this state, and is there a way to transform $$|C^n Z \rangle$$ to many $$|CCZ \rangle$$ magic states? If so, what is the rate w.r.t. $$n$$?

The simplest thing you could possibly do is to turn one $$|C^nZ\rangle$$ state into $$O(1/2^n)$$ $$|CCZ\rangle$$ states by measuring out $$n-3$$ of the qubits and discarding if any of the measurement results are $$|0\rangle$$ instead of $$|1\rangle$$.
At the moment I can't think of anything better. It seems like there should be some way of doing a conversion that has cost $$O(1/n)$$ instead of $$O(1/2^n)$$, probably involving catalysis, but nothing came to mind for how to actually do it.
If the $$|C^nZ\rangle$$ state was somehow magically unfolded into a $$W_{2^n}$$ state via a binary-to-unary converter at no cost, then you could deterministically recover one CCZ state by using measurement based uncomputation to uncompute CSWAP gates folding it back into a $$W_8$$ state followed by performing $$W_8 \rightarrow CCZ$$ which is 1-to-1.
For an upper bound, stabilizer nullity probably shows the rate can't be more than $$(n+1)/3$$. So it's somehwere between $$(n+1)/3$$ and $$4/2^n$$. Wide margins.
• Thanks! It's unfortunate that the bounds are so wide. What if I instead move the goalposts and say I can perform a FT $C^n Z$ gate? I wonder if I can somehow use this to perform $\Theta(n)$ $CCZ$ gates, with the idea of state catalysis also what I had in mind. Is it any clearer in that case? Commented Jul 9 at 20:35